Coinduction, the dual of induction, is a mathematical principle for reasoning about infinite and circular structures. Originally studied in the field of concurrency theory, by now it is evident that coinductive techniques are ubiquitous in computer science, mathematics and logic. In particular, coinduction has led to a new foundation of automata theory, in turn leading to novel algorithms and methods for various kinds of automata. These applications are driven by the modelling of automata as coalgebras, an abstract framework for the uniform study of dynamical systems. This course provides a gentle introduction to coinduction and coalgebras, using automata as a motivating example.